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In this video we will look at a couple of examples to practice thinking about conjectures and counter examples.

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Example A says: here's an algebraic equation and a table of values for <i>n</i> with the results for <i>t</i>.

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Notice that we started out with this equation:

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And what we have here is really just a table,

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that's divided in three parts, we have our values for <i>n</i>, all sort of calculations in the middle, and our answer fot <i>t</i>.

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And after looking at the table, Pablo makes this conjecture:

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the value of (<i>n</i>-1)(<i>n</i>-2)(<i>n</i>-3), in other words, the answer for <i>t</i> is zero for any hole number value of <i>n</i>.

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So, he is basically saying, that no matter what number I plug in over on the left from <i>n</i>,

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my answer is always going to end up being zero

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because that's what happen in the first three times. So maybe it will always happen.

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The question that we have is: is this a valid, or true, conjecture?

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So, if it's true, it means that it would be true for any number that you plug in for <i>n</i>.

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So you can plug in 100, and the answer should still be zero for <i>t</i>.

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So let's just test it out. Let's just try 100. Let's say <i>n</i>=100.

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We are trying to see if <i>t</i> will actually be equal zero.
So let's plug it in.

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We would have (100-1)(100-2)(100-3).

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100-1 is 99..

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Then we've got times 98,

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and times 97.

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Now I know that the answer to that is not zero,

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because to have an answer zero you have to multiply some of the one of the line by zero.

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So, this number is not equal zero. It's just gonna be a big number, definetly not equal to zero.

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So that means actually that his conjecture is not valid.


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It's not true and what I just did over here: <i>n</i> equals 100, that's a counter example.

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Because it's a specific example that shows that his conjecture is wrong.

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I can plug in, for example, 100 into that expression for <i>t</i> and my answer is not zero. Therefore, he is wrong.

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So a counter example is just one example to prove someone is wrong.

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That counter prefix means like going against the clean.

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Alright, let's look at example B:

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*Arthur is making figures for a graphic art project. He drew polygons and some of their diagonals.*

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And here we have four examples, based on these examples, Arthur made this conjecture:

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*if a convex polygon has <i>n</i> sides, then there are <i>n</i>-2 triangles drawn from any given vertex of the polygon.*

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So let's just think about what that means. He is saying that, if the shape has <i>n</i> sides,

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for example, this would be <i>n</i> equals 3, three sides,

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four sides,

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five sides,

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six sides.

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He is saying that there are always <i>n</i> minus 2 triangles.

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So, for example, if <i>n</i> is five, 5-2 is 3, he is saying, in this example, there are 3 triangles: one, two, three.

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And in the next one that are four triangles, which is 6-2.

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So, the question is: is Arthur's conjecture correct? Can you find a counter example to his conjecture?

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Well, his conjecture certainly appears to be correct based on the four examples that he has done.

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And we could do more examples and it would actually turn out that any example you'll try, it will be true,

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but you still haven't proven it, if you've just looked at examples, because it is still possible

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that that's another example out of there that you haven't thought out. That would be a counter example to his conjecture.

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So what we should say is:

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his conjecture appears to be true,

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but it still needs to be proven.

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Because just looking at examples isn't an official really proof for sure that it will always be true.